\(\int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx\) [147]

Optimal result

Integrand size = 19, antiderivative size = 275 \[ \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx=\frac {a d^2 x}{e^3}-\frac {b d x}{2 c e^2}+\frac {b x^2}{6 c e}+\frac {b d \text {arctanh}(c x)}{2 c^2 e^2}+\frac {b d^2 x \text {arctanh}(c x)}{e^3}-\frac {d x^2 (a+b \text {arctanh}(c x))}{2 e^2}+\frac {x^3 (a+b \text {arctanh}(c x))}{3 e}+\frac {d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{e^4}-\frac {d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e^4}+\frac {b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}+\frac {b \log \left (1-c^2 x^2\right )}{6 c^3 e}-\frac {b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 e^4}+\frac {b d^3 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e^4} \] Output:

a*d^2*x/e^3-1/2*b*d*x/c/e^2+1/6*b*x^2/c/e+1/2*b*d*arctanh(c*x)/c^2/e^2+b*d 
^2*x*arctanh(c*x)/e^3-1/2*d*x^2*(a+b*arctanh(c*x))/e^2+1/3*x^3*(a+b*arctan 
h(c*x))/e+d^3*(a+b*arctanh(c*x))*ln(2/(c*x+1))/e^4-d^3*(a+b*arctanh(c*x))* 
ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/e^4+1/2*b*d^2*ln(-c^2*x^2+1)/c/e^3+1/6*b*l 
n(-c^2*x^2+1)/c^3/e-1/2*b*d^3*polylog(2,1-2/(c*x+1))/e^4+1/2*b*d^3*polylog 
(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e^4
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.47 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.72 \[ \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx=\frac {-\frac {b e^3}{c^3}+6 a d^2 e x-\frac {3 b d e^2 x}{c}-3 a d e^2 x^2+\frac {b e^3 x^2}{c}+2 a e^3 x^3+\frac {3 b d e^2 \text {arctanh}(c x)}{c^2}-3 i b d^3 \pi \text {arctanh}(c x)+6 b d^2 e x \text {arctanh}(c x)-3 b d e^2 x^2 \text {arctanh}(c x)+2 b e^3 x^3 \text {arctanh}(c x)-6 b d^3 \text {arctanh}\left (\frac {c d}{e}\right ) \text {arctanh}(c x)+3 b d^3 \text {arctanh}(c x)^2-\frac {3 b d^2 e \text {arctanh}(c x)^2}{c}+\frac {3 b d^2 \sqrt {1-\frac {c^2 d^2}{e^2}} e e^{-\text {arctanh}\left (\frac {c d}{e}\right )} \text {arctanh}(c x)^2}{c}+6 b d^3 \text {arctanh}(c x) \log \left (1+e^{-2 \text {arctanh}(c x)}\right )+3 i b d^3 \pi \log \left (1+e^{2 \text {arctanh}(c x)}\right )-6 b d^3 \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-6 b d^3 \text {arctanh}(c x) \log \left (1-e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )-6 a d^3 \log (d+e x)+\frac {3 b d^2 e \log \left (1-c^2 x^2\right )}{c}+\frac {b e^3 \log \left (1-c^2 x^2\right )}{c^3}+\frac {3}{2} i b d^3 \pi \log \left (1-c^2 x^2\right )+6 b d^3 \text {arctanh}\left (\frac {c d}{e}\right ) \log \left (i \sinh \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )\right )-3 b d^3 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}(c x)}\right )+3 b d^3 \operatorname {PolyLog}\left (2,e^{-2 \left (\text {arctanh}\left (\frac {c d}{e}\right )+\text {arctanh}(c x)\right )}\right )}{6 e^4} \] Input:

Integrate[(x^3*(a + b*ArcTanh[c*x]))/(d + e*x),x]
 

Output:

(-((b*e^3)/c^3) + 6*a*d^2*e*x - (3*b*d*e^2*x)/c - 3*a*d*e^2*x^2 + (b*e^3*x 
^2)/c + 2*a*e^3*x^3 + (3*b*d*e^2*ArcTanh[c*x])/c^2 - (3*I)*b*d^3*Pi*ArcTan 
h[c*x] + 6*b*d^2*e*x*ArcTanh[c*x] - 3*b*d*e^2*x^2*ArcTanh[c*x] + 2*b*e^3*x 
^3*ArcTanh[c*x] - 6*b*d^3*ArcTanh[(c*d)/e]*ArcTanh[c*x] + 3*b*d^3*ArcTanh[ 
c*x]^2 - (3*b*d^2*e*ArcTanh[c*x]^2)/c + (3*b*d^2*Sqrt[1 - (c^2*d^2)/e^2]*e 
*ArcTanh[c*x]^2)/(c*E^ArcTanh[(c*d)/e]) + 6*b*d^3*ArcTanh[c*x]*Log[1 + E^( 
-2*ArcTanh[c*x])] + (3*I)*b*d^3*Pi*Log[1 + E^(2*ArcTanh[c*x])] - 6*b*d^3*A 
rcTanh[(c*d)/e]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 6*b*d^ 
3*ArcTanh[c*x]*Log[1 - E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))] - 6*a*d^3 
*Log[d + e*x] + (3*b*d^2*e*Log[1 - c^2*x^2])/c + (b*e^3*Log[1 - c^2*x^2])/ 
c^3 + ((3*I)/2)*b*d^3*Pi*Log[1 - c^2*x^2] + 6*b*d^3*ArcTanh[(c*d)/e]*Log[I 
*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]] - 3*b*d^3*PolyLog[2, -E^(-2*ArcTan 
h[c*x])] + 3*b*d^3*PolyLog[2, E^(-2*(ArcTanh[(c*d)/e] + ArcTanh[c*x]))])/( 
6*e^4)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6502, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx\)

\(\Big \downarrow \) 6502

\(\displaystyle \int \left (-\frac {d^3 (a+b \text {arctanh}(c x))}{e^3 (d+e x)}+\frac {d^2 (a+b \text {arctanh}(c x))}{e^3}-\frac {d x (a+b \text {arctanh}(c x))}{e^2}+\frac {x^2 (a+b \text {arctanh}(c x))}{e}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^3 \log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{e^4}-\frac {d^3 (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e^4}-\frac {d x^2 (a+b \text {arctanh}(c x))}{2 e^2}+\frac {x^3 (a+b \text {arctanh}(c x))}{3 e}+\frac {a d^2 x}{e^3}+\frac {b d \text {arctanh}(c x)}{2 c^2 e^2}+\frac {b d^2 x \text {arctanh}(c x)}{e^3}+\frac {b d^2 \log \left (1-c^2 x^2\right )}{2 c e^3}+\frac {b \log \left (1-c^2 x^2\right )}{6 c^3 e}-\frac {b d^3 \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 e^4}+\frac {b d^3 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e^4}-\frac {b d x}{2 c e^2}+\frac {b x^2}{6 c e}\)

Input:

Int[(x^3*(a + b*ArcTanh[c*x]))/(d + e*x),x]
 

Output:

(a*d^2*x)/e^3 - (b*d*x)/(2*c*e^2) + (b*x^2)/(6*c*e) + (b*d*ArcTanh[c*x])/( 
2*c^2*e^2) + (b*d^2*x*ArcTanh[c*x])/e^3 - (d*x^2*(a + b*ArcTanh[c*x]))/(2* 
e^2) + (x^3*(a + b*ArcTanh[c*x]))/(3*e) + (d^3*(a + b*ArcTanh[c*x])*Log[2/ 
(1 + c*x)])/e^4 - (d^3*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c*d + e) 
*(1 + c*x))])/e^4 + (b*d^2*Log[1 - c^2*x^2])/(2*c*e^3) + (b*Log[1 - c^2*x^ 
2])/(6*c^3*e) - (b*d^3*PolyLog[2, 1 - 2/(1 + c*x)])/(2*e^4) + (b*d^3*PolyL 
og[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e^4)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6502

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p, ( 
f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p, 0] 
 && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])
 

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.17

method	result	size
parts	\(\frac {a \,x^{3}}{3 e}-\frac {a d \,x^{2}}{2 e^{2}}+\frac {a \,d^{2} x}{e^{3}}-\frac {a \,d^{3} \ln \left (e x +d \right )}{e^{4}}+\frac {b \left (\frac {c^{4} \operatorname {arctanh}\left (c x \right ) x^{3}}{3 e}-\frac {c^{4} \operatorname {arctanh}\left (c x \right ) x^{2} d}{2 e^{2}}+\frac {c^{4} \operatorname {arctanh}\left (c x \right ) x \,d^{2}}{e^{3}}-\frac {c^{4} \operatorname {arctanh}\left (c x \right ) d^{3} \ln \left (c e x +c d \right )}{e^{4}}-\frac {c \left (\frac {c^{3} d^{3} \left (\frac {\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )}{2 e}-\frac {\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )}{2 e}\right )}{e^{2}}-\frac {-5 c d \left (c e x +c d \right )+\left (c e x +c d \right )^{2}+\left (3 c^{2} d^{2}-\frac {3}{2} c d e +e^{2}\right ) \ln \left (c e x -e \right )+\left (3 c^{2} d^{2}+\frac {3}{2} c d e +e^{2}\right ) \ln \left (c e x +e \right )}{6 e^{2}}\right )}{e}\right )}{c^{4}}\)	\(323\)
derivativedivides	\(\frac {\frac {a \,c^{4} d^{2} x}{e^{3}}-\frac {a \,c^{4} d \,x^{2}}{2 e^{2}}+\frac {a \,c^{4} x^{3}}{3 e}-\frac {a \,c^{4} d^{3} \ln \left (c e x +c d \right )}{e^{4}}+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} x}{e^{3}}-\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d \,x^{2}}{2 e^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3 e}-\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d^{3} \ln \left (c e x +c d \right )}{e^{4}}-\frac {\frac {c^{3} d^{3} \left (\frac {\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )}{2 e}-\frac {\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )}{2 e}\right )}{e^{2}}+\frac {5 c d \left (c e x +c d \right )-\left (c e x +c d \right )^{2}+\left (-3 c^{2} d^{2}+\frac {3}{2} c d e -e^{2}\right ) \ln \left (-c e x +e \right )+\left (-3 c^{2} d^{2}-\frac {3}{2} c d e -e^{2}\right ) \ln \left (-c e x -e \right )}{6 e^{2}}}{e}\right )}{c^{4}}\)	\(347\)
default	\(\frac {\frac {a \,c^{4} d^{2} x}{e^{3}}-\frac {a \,c^{4} d \,x^{2}}{2 e^{2}}+\frac {a \,c^{4} x^{3}}{3 e}-\frac {a \,c^{4} d^{3} \ln \left (c e x +c d \right )}{e^{4}}+b c \left (\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d^{2} x}{e^{3}}-\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d \,x^{2}}{2 e^{2}}+\frac {\operatorname {arctanh}\left (c x \right ) c^{3} x^{3}}{3 e}-\frac {\operatorname {arctanh}\left (c x \right ) c^{3} d^{3} \ln \left (c e x +c d \right )}{e^{4}}-\frac {\frac {c^{3} d^{3} \left (\frac {\operatorname {dilog}\left (\frac {c e x -e}{-c d -e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x -e}{-c d -e}\right )}{2 e}-\frac {\operatorname {dilog}\left (\frac {c e x +e}{-c d +e}\right )+\ln \left (c e x +c d \right ) \ln \left (\frac {c e x +e}{-c d +e}\right )}{2 e}\right )}{e^{2}}+\frac {5 c d \left (c e x +c d \right )-\left (c e x +c d \right )^{2}+\left (-3 c^{2} d^{2}+\frac {3}{2} c d e -e^{2}\right ) \ln \left (-c e x +e \right )+\left (-3 c^{2} d^{2}-\frac {3}{2} c d e -e^{2}\right ) \ln \left (-c e x -e \right )}{6 e^{2}}}{e}\right )}{c^{4}}\)	\(347\)
risch	\(-\frac {b d \ln \left (c x +1\right ) x^{2}}{4 e^{2}}+\frac {b d \ln \left (c x +1\right )}{4 c^{2} e^{2}}+\frac {b \ln \left (c x +1\right ) x \,d^{2}}{2 e^{3}}+\frac {b \ln \left (c x +1\right ) d^{2}}{2 c \,e^{3}}-\frac {b \,d^{3} \ln \left (c x +1\right ) \ln \left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 e^{4}}-\frac {b d \ln \left (-c x +1\right )}{4 c^{2} e^{2}}-\frac {b \ln \left (-c x +1\right ) x \,d^{2}}{2 e^{3}}+\frac {b d \ln \left (-c x +1\right ) x^{2}}{4 e^{2}}+\frac {b \ln \left (-c x +1\right ) d^{2}}{2 c \,e^{3}}+\frac {b \,d^{3} \ln \left (-c x +1\right ) \ln \left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 e^{4}}-\frac {b d x}{2 c \,e^{2}}+\frac {b \,x^{2}}{6 c e}-\frac {11 b}{18 c^{3} e}-\frac {a}{3 c^{3} e}+\frac {a \,x^{3}}{3 e}+\frac {a \,d^{2} x}{e^{3}}-\frac {b \,d^{3} \operatorname {dilog}\left (\frac {\left (c x +1\right ) e +c d -e}{c d -e}\right )}{2 e^{4}}+\frac {b \ln \left (c x +1\right )}{6 c^{3} e}+\frac {b \ln \left (c x +1\right ) x^{3}}{6 e}-\frac {b \,d^{2}}{c \,e^{3}}+\frac {a d}{2 c^{2} e^{2}}-\frac {a \,d^{2}}{c \,e^{3}}-\frac {b \ln \left (-c x +1\right ) x^{3}}{6 e}+\frac {b \ln \left (-c x +1\right )}{6 c^{3} e}+\frac {b \,d^{3} \operatorname {dilog}\left (\frac {\left (-c x +1\right ) e -c d -e}{-c d -e}\right )}{2 e^{4}}-\frac {a d \,x^{2}}{2 e^{2}}-\frac {a \,d^{3} \ln \left (\left (-c x +1\right ) e -c d -e \right )}{e^{4}}\)	\(484\)

Input:

int(x^3*(a+b*arctanh(c*x))/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/3*a/e*x^3-1/2*a/e^2*d*x^2+a*d^2*x/e^3-a*d^3/e^4*ln(e*x+d)+b/c^4*(1/3*c^4 
*arctanh(c*x)/e*x^3-1/2*c^4*arctanh(c*x)/e^2*x^2*d+c^4*arctanh(c*x)/e^3*x* 
d^2-c^4*arctanh(c*x)*d^3/e^4*ln(c*e*x+c*d)-c/e*(1/e^2*c^3*d^3*(1/2/e*(dilo 
g((c*e*x-e)/(-c*d-e))+ln(c*e*x+c*d)*ln((c*e*x-e)/(-c*d-e)))-1/2/e*(dilog(( 
c*e*x+e)/(-c*d+e))+ln(c*e*x+c*d)*ln((c*e*x+e)/(-c*d+e))))-1/6/e^2*(-5*c*d* 
(c*e*x+c*d)+(c*e*x+c*d)^2+(3*c^2*d^2-3/2*c*d*e+e^2)*ln(c*e*x-e)+(3*c^2*d^2 
+3/2*c*d*e+e^2)*ln(c*e*x+e))))
 

Fricas [F]

\[ \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{3}}{e x + d} \,d x } \] Input:

integrate(x^3*(a+b*arctanh(c*x))/(e*x+d),x, algorithm="fricas")
 

Output:

integral((b*x^3*arctanh(c*x) + a*x^3)/(e*x + d), x)
 

Sympy [F]

\[ \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int \frac {x^{3} \left (a + b \operatorname {atanh}{\left (c x \right )}\right )}{d + e x}\, dx \] Input:

integrate(x**3*(a+b*atanh(c*x))/(e*x+d),x)
 

Output:

Integral(x**3*(a + b*atanh(c*x))/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{3}}{e x + d} \,d x } \] Input:

integrate(x^3*(a+b*arctanh(c*x))/(e*x+d),x, algorithm="maxima")
 

Output:

-1/6*a*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3) + 
1/2*b*integrate(x^3*(log(c*x + 1) - log(-c*x + 1))/(e*x + d), x)
 

Giac [F]

\[ \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )} x^{3}}{e x + d} \,d x } \] Input:

integrate(x^3*(a+b*arctanh(c*x))/(e*x+d),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x) + a)*x^3/(e*x + d), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}{d+e\,x} \,d x \] Input:

int((x^3*(a + b*atanh(c*x)))/(d + e*x),x)
 

Output:

int((x^3*(a + b*atanh(c*x)))/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {x^3 (a+b \text {arctanh}(c x))}{d+e x} \, dx=\frac {6 \left (\int \frac {\mathit {atanh} \left (c x \right ) x^{3}}{e x +d}d x \right ) b \,e^{4}-6 \,\mathrm {log}\left (e x +d \right ) a \,d^{3}+6 a \,d^{2} e x -3 a d \,e^{2} x^{2}+2 a \,e^{3} x^{3}}{6 e^{4}} \] Input:

int(x^3*(a+b*atanh(c*x))/(e*x+d),x)
 

Output:

(6*int((atanh(c*x)*x**3)/(d + e*x),x)*b*e**4 - 6*log(d + e*x)*a*d**3 + 6*a 
*d**2*e*x - 3*a*d*e**2*x**2 + 2*a*e**3*x**3)/(6*e**4)